Oberwolfach Seminar: Algebraic Properties of Profinite Groups

Date

June 12^th - June 18^th, 2011

Organizers

Nikolay Nikolov, London

Dan Segal, Oxford

ID

1124a

Programme

An introduction to profinite groups from an algebraic (group-theoretic) point of view. The aims are

1. to explain various methods that have been used recently to illuminate the algebraic properties of finitely generated profinite groups,
2. to explore the consequences, both for profnite groups and for infinite group theory in general, and
3. to explore which profinite groups can be the profinite completions of finitely generated (abstract) groups.

Major results to be covered (in the lectures or in the examples classes) will include the solution of Serre's problem on finite-index subgroups in finitely generated profinite groups (due to Nikolov and Segal); related results about verbal width in pro-p groups (due to Jaikin-Zapirain); the theorem of Jarden and Lubotzky on elementary equivalence of profinite groups; The theorem of Nori and Weisfeiler on strong approximation for linear groups, and its application to polynomial subgroup growth groups by Lubotzky and Mann.

Not all will be proved in full: the emphasis will be on understanding the underlying ideas and exploring the implications.

Preparatory reading

In most cases, it will suffice to understand the main definitions and the statements of the main theorems: you are not expected to work through all the proofs.

• Any standard text on group theory (e.g. D. J. S. Robinson's 'course');
• Profinite Groups by J. S. Wilson: Chapter 1;
• Analytic pro-p Groups by Dixon, du Sautoy, Mann and Segal: Prelude and Chapter 1;
• Finite Simple Groups by R. A. Wilson: Chapter 1; main definitions in Chapters 2 - 4;
• Infinite Linear Groups by B. A. F. Wehrfriz: Chapters 1 - 4;
• Strong Approximation Methods in Group Theory by N. Nikolov, arXiv:0803.4165;
• Subgroup Growth by Lubotzky and Segal: Windows 1, 2, 4, 7, 8, 9.

Deadline for applications

April 15^th, 2011

The seminars take place at the Mathematisches Forschungsinstitut Oberwolfach. The number of participants is restricted to 25. The Institute covers accommodation and food. We are pleased that the Carl Friedrich von Siemens Foundation has decided to support the Oberwolfach Seminars from summer 2008 to summer 2013. By this support, travel expenses can be reimbursed up to 200 Euro in average per person. Participants can ask for travel support during their stay in Oberwolfach at the guest office against copy of travel receipts. Applications including

• full name and address, including e-mail address
• present position, university
• name of supervisor of Ph.D. thesis
• a short summary of previous work and interest

should be sent preferably by e-mail (.ps or .pdf file) to:

    Prof. Dr. Dr. h.c. Gert-Martin Greuel
    Universität Kaiserslautern
    Fachbereich Mathematik
    Erwin Schrödingerstr.
    67663 Kaiserslautern, Germany